1. Field of the Invention
The present invention relates generally to marine seismic surveying and, more particularly, to a method for attenuating the effect of surface multiples in a marine seismic signal.
2. Description of the Related Art
Seismic surveying is a method for determining the structure of subterranean formations in the earth. Seismic surveying typically utilizes seismic energy sources which generate seismic waves and seismic receivers which detect seismic waves. The seismic waves propagate into the formations in the earth, where a portion of the waves reflects from interfaces between subterranean formations. The amplitude and polarity of the reflected waves are determined by the differences in acoustic impedance between the rock layers comprising the subterranean formations. The acoustic impedance of a rock layer is the product of the acoustic propagation velocity within the layer and the density of the layer. The seismic receivers detect the reflected seismic waves and convert the reflected waves into representative electrical signals. The signals are typically transmitted by electrical, optical, radio or other means to devices which record the signals. Through analysis of the recorded signals, the shape, position and composition of the subterranean formations can be determined.
Marine seismic surveying is a method for determining the structure of subterranean formations underlying bodies of water. Marine seismic surveying typically utilizes seismic energy sources and seismic receivers located in the water which are either towed behind a vessel or positioned on the water bottom from a vessel. The energy source is typically an explosive device or compressed air system which generates seismic energy, which then propagates as seismic waves through the body of water and into the earth formations below the bottom of the water. As the seismic waves strike interfaces between subterranean formations, a portion of the seismic waves reflects back through the earth and water to the seismic receivers, to be detected, transmitted, and recorded. The seismic receivers typically used in marine seismic surveying are pressure sensors, such as hydrophones. Additionally, though, motion sensors, such as geophones or accelerometers may be used. Both the sources and receivers may be strategically repositioned to cover the survey area.
Seismic waves, however, reflect from interfaces other than just those between subterranean formations, as would be desired. Seismic waves also reflect from the water bottom and the water surface, and the resulting reflected waves themselves continue to reflect. Waves which reflect multiple times are called "multiples". Waves which reflect multiple times in the water layer between the water surface above and the water bottom below are called "water-bottom multiples". Water-bottom multiples have long been recognized as a problem in marine seismic processing and interpretation, so multiple attenuation methods based on the wave equation have been developed to handle water-bottom multiples. However, a larger set of multiples containing water-bottom multiples as a subset can be defined. The larger set includes multiples with upward reflections from interfaces between subterranean formations in addition to upward reflections from the water bottom. The multiples in the larger set have in common their downward reflections at the water surface and thus are called "surface multiples". FIG. 1, discussed below, provides examples of different types of reflections.
FIG. 1 shows a diagrammatic view of marine seismic surveying. The procedure is designated generally as 100. Subterranean formations to be explored, such as 102 and 104, lie below a body of water 106. Seismic energy sources 108 and seismic receivers 110 are positioned in the body of water 106, typically by one or more seismic vessels (not shown). A seismic source 108, such as an air gun, creates seismic waves in the body of water 106 and a portion of the seismic waves travels downward through the water toward the subterranean formations 102 and 104 beneath the body of water 106. When the seismic waves reach a seismic reflector, a portion of the seismic waves reflects upward and a portion of the seismic waves continues downward. The seismic reflector can be the water bottom 112 or one of the interfaces between subterranean formation, such as interface 114 between formations 102 and 104. When the reflected waves travelling upward reach the water/air interface at the water surface 116, a majority portion of the waves reflects downward again. Continuing in this fashion, seismic waves can reflect multiple times between upward reflectors, such as the water bottom 112 or formation interfaces below, and the downward reflector at the water surface 116 above, as described more fully below. Each time the reflected waves propagate past the position of a seismic receiver 110, the receiver 110 senses the reflected waves and generates representative signals.
Primary reflections are those seismic waves which have reflected only once, from the water bottom 112 or an interface between subterranean formations, before being detected by a seismic receiver 110. An example of a primary reflection is shown in FIG. 1 by raypaths 120 and 122. Primary reflections contain the desired information about the subterranean formations which is the goal of marine seismic surveying. Surface multiples are those waves which have reflected multiple times between the water surface 116 and any upward reflectors, such as the water bottom 112 or formation interfaces, before being sensed by a receiver 110. An example of a surface multiple which is specifically a water bottom multiple is shown by raypaths 130, 132, 134 and 136. The surface multiple starting at raypath 130 is a multiple of order one, since the multiple contains one reflection from the water surface 116. Two examples of general surface multiples with upward reflections from both the water bottom 112 and formation interfaces are shown by raypaths 140, 142, 144, 146, 148 and 150 and by raypaths 160, 162, 164, 166, 168 and 170. Both of these latter two examples of surface multiples are multiples of order two, since the multiples contain two reflections from the water surface 116. In general, a surface multiple is of order i if the multiple contains i reflections from the water surface 116. Surface multiples are extraneous noise which obscures the desired primary reflection signal.
Surface multiple attenuation is a prestack inversion of a recorded wavefield which removes all orders of all surface multiples present within the marine seismic signal. Unlike some wave-equation-based multiple-attenuation algorithms, surface multiple attenuation does not require any modeling of or assumptions regarding the positions, shapes and reflection coefficients of the multiple-causing reflectors. Instead, surface multiple attenuation relies on the internal physical consistency between primary and multiple events that must exist in any properly recorded marine data set. The information needed for the surface multiple attenuation process is already contained within the seismic data.
In the following discussion, let the original seismic wavefields, the corresponding recorded data sets, or the corresponding data cubes or matrices be represented by the same upper-case letters. Thus let D represent a marine seismic data set corresponding to a wavefield D. The wavefield D can be divided into two parts, EQU D=P+M (1)
The primary wavefield, P, represents that portion of D which contains no surface multiples. The surface multiples wavefield, M, represents that portion of D which contains surface multiples of any order. Surface multiple attenuation is a processing method for removing the multiples wavefield M from the recorded wavefield D to yield the desired primary wavefield P.
For each i from 1 to .infin., let M.sub.i represent that portion of M containing surface multiples of order i. Then the surface multiple wavefield M can be further decomposed into an infinite sum of different orders, EQU M=M.sub.1 +M.sub.2 + . . . +M.sub.i + . . . (2)
Recorded data sets have a finite duration, so only a finite number of terms from Eq. (2) are needed to represent the corresponding wavefield. Substituting an appropriately truncated Eq. (2) into Eq. (1) yields EQU D=P+M.sub.1 +M.sub.2 + . . . +M.sub.n, (3)
for some value n.
The process of surface multiple attenuation assumes that surface multiple events M.sub.i of order i can be predicted from knowledge of both the surface multiple events M.sub.i-1 of order i-1 and the primary wavefield P. This assumption means that there exists some mathematical operator O such that EQU M.sub.i =POM.sub.i-1. (4)
Inserting Eq. (4) into Eq. (3) and factoring out first P and then O yields ##EQU1## Define a truncated version D.sub.T of D by ##EQU2## In practice, as will be discussed later, D.sub.T would be approximated by truncating the seismic traces in D in time rather than actually constructing and subtracting M.sub.n from D. Inserting Eq. (6) into Eq. (5) yields the compact form D=P[1+OD.sub.T ] (7)
Eq. (7) is a formula for recursive forward modeling of surface multiples. Eq. (7) represents adding the events of order n to the wavefield containing all events up to and including order n-1. If the bracketed expression in Eq. (7) has an inverse, then Eq. (7) can be inverted to yield EQU P=D[1+OD.sub.T ].sup.-1. (8)
Eq. (8) is the inverse of the recursive forward modeling equation, Eq. (7). Eq. (8) states that if a suitable operator O can be found, then the primary wavefield P, free of surface multiples, can be computed directly from the recorded wavefield D. The operator O being suitable means that the operator O must be both geophysically and mathematically plausible. The operator O being geophysically plausible means that the operator O satisfies Eq. (4). The operator O being mathematically plausible means firstly that the factorizations in Eq. (5) are valid and secondly that the inverse of the bracketed expression in Eq. (7) exists and thus Eq. (8) is valid. The Kirchhoff integral, a mathematical statement of Huygens' principle, honors the wave equation and can be used as the basis of a geophysically suitable operator O.
Use of the Kirchhoff integral provides the appropriate two- or three-dimensional generalization of the inverse of the recursive forward modeling equation for surface multiple attenuation, as given by Eq. (8). The Kirchhoff integral must be made compatible with Eqs. (4) through (8). First, the recorded marine seismic data are Fourier transformed from the time domain to the frequency domain. Let individual seismic traces or events within the wavefields or data sets be represented by lower-case letters. Thus m.sub.i, is a multiple event of order i within a seismic trace d in the wavefield D. Let p and m represent single-frequency components of Fourier-transformed seismic traces. For example, m.sub.i (s,r) is one frequency component of the seismic trace whose source and receiver were at positions s and r, respectively, and which contains only surface multiples of order i. Let m.sub.M,i-1 represent m.sub.-1 after being modified to include the scale and phase corrections and the obliquity factor required by the Kirchhoff integral. The Kirchhoff modification is given by ##EQU3## where x=inline coordinate,
j=(-1).sup.1/2 PA1 .omega.=angular frequency, PA1 k.sub.x =x-component of wavenumber vector, and PA1 V=speed of sound in water.
Because of k.sub.x, the modification of m.sub.i-1 is dip-dependent. In the frequency domain, the Kirchhoff integral can be written as EQU m.sub.i (s,r)=-.intg.p(s,x)m.sub.M,i-1 (x,r)dx. (10)
The minus sign is due to the negative reflection coefficient of the water surface.
In practice, recorded wavefields are not continuous in x, so the Kirchhoff integral in Eq. (10) has to be replaced by the following discrete summation over x EQU m.sub.i (s,r)=-.SIGMA.p(s,x)m.sub.M,i-1 (x,r). (11)
Except for the minus sign, Eq. (11) is the formula for computing an element of the product of two matrices. Thus, define matrix M.sub.i-1, as the matrix whose columns are the common-receiver records, m.sub.i-1 (x,r), define matrix M.sub.M,i-1 as the matrix whose columns are the Kirchhoff-modified common-receiver records, m.sub.M,i-1 (x,r), and define matrix P as the matrix whose rows are the common-shot records, p(s,x). Then Eq. (11) becomes EQU M.sub.i =PM.sub.M,i-1. (12)
Since the matrix indices are the shot and receiver coordinates, the zero-offset seismic traces lie along the main diagonal of each matrix. If the operator O in Eq. (4) is matrix multiplication and the quantities in uppercase are matrices, then Eq. (4) becomes Eq. (12) and so Eq. (8) becomes EQU P=D[I-D.sub.M ].sup.-1, (13)
where I is the identity matrix and the "-1" superscript indicates matrix inversion.
Eq. (13) provides a simple algorithm for two-dimensional surface multiple attenuation for ideal data. By "ideal" is meant that the wavefield is recorded broadband, contains no noise, has all wavelet effects, including source and receiver ghosts, removed, and has a trace-offset range that begins at zero offset. Furthermore, each individual sample within the ideal data set D must have a true relative amplitude with respect to every other sample within D. For non-ideal data, wavelet effects have not been removed and are the major factor which must be taken into consideration. In theory, this problem is easily fixed by redefining operator O to include a convolution by the inverse of the source wavelet w. Eq. (13) becomes EQU P=D[I-w.sup.-1 D.sub.M ].sup.-1, (14)
where w.sup.-1 is the wavelet inverse. Since Eq. (14) is in the frequency domain, convolution is accomplished by multiplication.
In practice, however, the source wavelet w is initially unknown. The source wavelet w can be found by minimizing the total energy in P in Eq. (14). Thus surface multiple attenuation becomes an L.sub.2 -norm minimization problem, which has standard methods of finding the solution, such as the conjugate gradient technique. One could also minimize other measures of the surface multiple energy in P.
The computation of P as described above requires computing the source wavelet w by minimizing the total energy in P in an iterative loop. This requires computing the inverse of the matrix quantity in brackets in Eq. (14) during each iteration for a new value of w. Computing the inverse of the matrix may be accomplished by a series expansion or an exact matrix inversion. The series expansion approach is approximate and can be slowly converging. Previous exact approaches to surface multiple attenuation require a large number of inversions of large square matrices. Square matrix inversion is computationally expensive.
U.S. Pat. No. 5,587,965 to Dragoset & Jericevic, titled "Surface Multiple Attenuation Via Eigenvalue Decomposition" describes a method using eigenvalue decomposition in an exact approach to solving Eq. (14) which is specially designed to make computing the matrix inverse in each iteration much less costly. The eigenvalue decomposition of matrix D.sub.M is computed, yielding EQU D.sub.M =S.SIGMA.S.sup.-1. (15)
Here .LAMBDA. is the diagonal matrix whose diagonal elements are the eigenvalues .lambda..sub.i of D.sub.M, that is, .LAMBDA.=diag [.lambda..sub.i ] . S is the square matrix whose columns are the corresponding eigenvectors of D.sub.M and S.sup.-1 is the matrix inverse of S. Inserting Eq. (15) into Eq. (14) yields EQU P=D[I-w.sup.-1 S .LAMBDA.S.sup.-1 ].sup.-1. (16)
Substituting I=S S.sup.-1 and factoring S and S.sup.-1 out to the left and right, respectively, allows one to rewrite Eq. (16) as EQU P=D S [I-w.sup.-1 .LAMBDA.].sup.-1 S.sup.-1. (17)
Now the expression in brackets in Eq. (17) is a diagonal matrix, since both I and .LAMBDA. are diagonal matrices and w.sup.-1 is a complex scalar. Thus Eq. (17) can also be written as EQU P=D S diag [1/(I-w.sup.-1 .lambda..sub.i)]S.sup.-1.
The expression in brackets in Eq. (17) can be inverted many times at low cost.
This derivation is described in U.S. Patent No. 5,587,965 to Dragoset & Jericevic. In their method, the matrix product D S and the inverse S.sup.-1 are computed only once and then saved in memory. After that initial cost, the incremental cost of recomputing Eq. (17) is that of a matrix multiplication and a diagonal matrix inversion, whereas the incremental cost of recomputing Eq. (14) is that of a matrix multiplication and a square matrix inversion. If the dimension of the square matrices involved is N, then recomputing Eq. (14) takes approximately 2N.sup.3 order of magnitude floating point operations for each iteration, while recomputing Eq. (17) takes only approximately (N.sup.3 +N) order of magnitude operations for each iteration but the first.